A Fixed-Point Iteration Method for High Frequency Helmholtz Equations

For numerically solving the high frequency Helmholtz equation, the conventional finite difference and finite element methods based on discretizing the equation on meshes usually suffer from the numerical dispersion errors ('pollution effect') that require very refined meshes (Babuska and Sauter in SIAM Rev 42(3):451-484, 2000). Asymptotic methods like geometrical optics provide an alternative way to compute the solutions without 'pollution effect', but they generally can only compute locally valid approximations for the solutions and fail to capture the caustics faithfully. In order to obtain globally valid solutions efficiently without 'pollution effect', we transfer the problem into a fixed-point problem related to an exponential operator, and the associated functional evaluations are achieved by unconditionally stable operator-splitting based pseudospectral schemes such that large step sizes are allowed to reach the approximated fixed point efficiently for certain prescribed accuracy requirement. And the Anderson acceleration is incorporated to accelerate the convergence. Both two-dimensional and three-dimensional numerical experiments are presented to demonstrate the method.